A stochastic, or random, process describes the correlation or evolution of random events. It is used to model stock market fluctuations and electronic/audio-visual/biological signals. Among the most well-known stochastic processes are random walks and Brownian motion.

learn more… | top users | synonyms

0
votes
1answer
16 views

Prove that the following process is a Geometric Brownian motion for every constant

Having some trouble understanding this problem: Given the dynamics of the geometric brownian motion $X_t$ where $(B_t)_{t\in\mathbf{R}_{+}}$ $$ dX_t = X_tdt+X_t dB_t,$$ $$X_0=1$$ for which value of ...
2
votes
1answer
47 views

Independence of a Stochastic Process at Distinct Time

Suppose $X_t$ is a stochastic process of $t$ on $[0,\infty)$ with almost surely continuous sample path. I have modified my question to the following one, per Math1000's comment below: Is the ...
0
votes
0answers
10 views

Memory less property of a Markov chain- Validation methods

Are there any tests to check the memory less property of a discrete time homogeneous Markov chain? I found a chi squared test to verify the time homogeneity of a Markov chain constructed from a set of ...
2
votes
1answer
29 views

How to compute $E[W_t^4]$, with $W_t$ being a standard Wiener process

I want to compute the fourth moment of a standard Wiener process: $E[W_t^4]$. My solution is not equal to the one in my textbook but I don't understand where I am wrong. I started by applying Ito's ...
0
votes
0answers
7 views

Markov Chains: Expected Return Time (Stochastic Process)

I am given a matrix with space {0,1,2,3,4}. I already calculated the invariable probability vector. However, the question asks to give the expected number of steps: -given Xo=0 to go back to state ...
0
votes
1answer
30 views

Deriving the Doob Meyer decomposition of a Sub Martingale using Ito's

Given the standard brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$ and defining the sub-m.g.: $$X_t =B^6_t+2t$$ I would like to derive its Doob-Meyer decomposition: [Sub-m.g.]= [increasing ...
0
votes
0answers
25 views

Size of families: Birth death immigration

The context of this problem is as follows. Starting from a population size of zero, immigrants arrive according to a homogeneous Poisson process with rate $\theta$. Once they arrive, immigrants start ...
0
votes
0answers
16 views

Quadratic variation question

Let $M$ be a vector of local martingales. Then there exist an increasing and adapted $C$ and optional processes $\sigma^{ij}, i,j=1,...,d$ such that $<M^i,M^j> = \int_0^. \sigma^{ij} dC_s$. Can ...
0
votes
0answers
16 views

Stochastic Process

I would like to know if anyone here could help me with this exercise. Y(t) = X(t +d) - X(t), where X(t) is a Gaussian Stochastic process. (A) Calculate the mean and covariance of Y(t) (B) Calculate ...
0
votes
0answers
23 views

Entry time and hitting time

Hi I have a question about entry time and hitting time. Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_{t})_{t \in[0,\infty)}$ be a $\mathbb{R}$-valued stochastic process on $(\Omega, ...
0
votes
0answers
11 views

Definition of mth order stationarity

in the definition of the weak GARCH processes they use the terminology of the 4th-order stationarity of the process $(X_t)$. I know the definition of 2n-order stationarity, but I'm not exactly sure, ...
2
votes
0answers
15 views

Convergence in distribution of stochastic equation solutions

I'm studying from Kurtz's book "Markov Processes Characterization and convergence" and I have a question about the convergence of processes in $\mathbb{Z}^d$ that are solution of some equation. (see ...
0
votes
1answer
39 views

Show martingale space is a Banach space

Let $\mathcal{H}^1 = \{M \in \mathcal{M}, E[sup_{t\geq 0} |M_t|] < \infty\}$, where $\mathcal{M}$ is the space of right continuous with left limits martingales. Show that $\mathcal{H}^1$ is ...
0
votes
1answer
19 views

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$

Suppose $X$ and $Y$ are independent exponential random variables with the same mean $µ = 1/2$. Let ($Z,W) := (X,X +Y)$ i) Find the regions where the joint pdf of $(Z,W)$ is positive. ii) Find the ...
1
vote
0answers
59 views

integral with respect to the point measure [on hold]

We have integral $$\int_0^tf(t-u)dX(u)$$ where $X(u)$ is random point process( or at least renewal process). Also it is known that $f(t)\sim t^{-\alpha},$ $0<\alpha<1$ as $t\rightarrow \infty$. ...
0
votes
0answers
14 views

stochastic process involving cdf of a process [on hold]

I would like to know if anyone here could help me out with this exercise. Here it goes: A stochastic process is created from Yn = c(n)Xn, where Xn is a stochastic process with mean equals to zero, ...
0
votes
0answers
23 views

Almost sure convergence of stochastic integral

Consider a probability space $(\Omega, \mathcal{F}, P)$ equipped with a Brownian motion $W$. Let $(\xi_n)_{n=1}^\infty$ be a sequence of adapted $\mathcal{F}(t)$-progressively measurable processes ...
0
votes
0answers
10 views

Aggregation Urn Distribution

I am trying to identify this distribution in terms of the number of balls, $n$, urns, $m$, and iterations, $i$. Before the first iteration each ball is independent. The first iteration consists of ...
-1
votes
1answer
35 views

Meaning of $\mathcal A_{\tau}$ for stopping time $\tau$.

Let $(X_n)$ be a stochastic process, adapted to a filtration $\mathcal A_n$, and let $\tau$ be a stopping time, then $$ \mathcal A_{\tau} := \left\{ A \in \sigma\left(\bigcup_n A_n\right) : A \cap \{ ...
1
vote
1answer
21 views

Expected Value of the exponential of a stochastic integral

What is the expected value of the following process: $$ e^{\int_0^t B_u\, du} $$ Thanks.
-1
votes
2answers
30 views

Brownian motion: first-hitting-time with double barrier [on hold]

Let $(B_t)_t$ be a standard ($B_0=0$) Brownian motion , and $$ T_{a,b} = \inf\{t>0 : B_t \not\in(a,b)\} $$ where $a<0<b$. What is the expected first-passage time $\mathbf{E}[T_{a,b}]$?
0
votes
2answers
20 views

Find the value of the real $\alpha$ for which exp($2B(t) - \alpha t$) is a martingale.

I tried to answer using the three conditions to be a martingale (measurability, integrability, and martingality), validating the integrability condition, which is $$ E |e^{2B(t) - \alpha t} | < ...
0
votes
1answer
19 views

What's the meaning of the state space with locally compact topological space?

I have encountered a statement in one paper describing the continuous-time controlled Markov chain with space state which is locally compact topological space. What does this mean? In my previous ...
1
vote
0answers
19 views

Deterministic integrals involving a Brownian motion [duplicate]

I am trying to work out the following two integrals involving a standard Brownian motion started at $W_0 = 0$. The first expression is bewildering me a bit, since it seems like somehow the Itô ...
2
votes
1answer
25 views

Pricing a riskless asset in the Black & Scholes market

Consider a Black&Scholes Market where a risky asset evolves according to: $$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$ $$S_o=s$$ Riskless asset is associated with risk free rate r. I want to represent ...
1
vote
0answers
36 views

Expected value and Variance of a stochastic time integral of a deterministic variable (Standard Brownian motion)

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, define: $$E(e^{\int_0^tudB_u})=?$$ $$ Var(e^{\int_0^tudB_u})=?$$ Sidenote to be edited later: Here is my try, I'm not capable to ...
0
votes
0answers
14 views

Need a little bit of guidance with stochastic processes

Let $X(t) = \begin{bmatrix} cos(t) + N(t)\\ sin(t) + S(t)\\ \end{bmatrix} $ (where $N(t)$ is a gaussian process and S(t) is a Poisson's process). Let ...
1
vote
0answers
67 views

Expected value of a brownian motion times the deterministic integral of a brownian motion

Given a Standard Brownian motion $(B_t)_{t\in\mathbf{R}_{+}}$, $E (B_t \int_0^tB_s^3ds)$ = ? I try to turn the expected value into a double integral by rewriting the $B_t$ term as 1) $E(\int_0^t ...
1
vote
0answers
24 views

Solve the stochastic differential equation

I have to solve the following SDE: $$dX_t=X_t dt+2W_tdW_t$$ Let $Y_t=X_t e^{-t}$. By Ito formula we have: $$dY_t=-X_te^{-t}dt+e^{-t}(X_t dt+2W_tdW_t)=2e^{-t}W_tdW_t$$ Thus ...
1
vote
0answers
43 views

Expected value of an exponential of a gaussian random variable

$$E (Y_t)=E(e^{X_t}) = E(e^{N(X_0e^{at};\frac{b^2}{2a}(e^{2at}-1)}) =\text{ ?}$$ Knowning that $$X_t \sim N\left[X_0e^{at};\frac{b^2}{2a}(e^{2at}-1)\right]$$ $$X_t= aX_t \, dt+b \, dB_t$$ The ...
0
votes
1answer
36 views

Proof that there exists a non-negative eigenvector corresponding to eigenvalue 1 of stochastic matrix

Let $P \in [0,1]^{n \times n}$ be a [irreducible or reducible] stochastic matrix where its rows sum to 1 i.e. $$ \forall i \in \{ 1 , \dots n \} \quad \sum_{j=1}^{n} P_{ij} = 1 $$ It is easy to show ...
0
votes
0answers
18 views

Definition of Simple Predictable Process

I am reading Protter's book "Stochastic Integration and Differential Equations". He (page 51) defines $H$ to be a simple predictable processes if it has a representation ...
1
vote
0answers
16 views

About moments in a quantile processes

Let $q_{n}(t)$ be the $nth$ quantile processes ($t\in (0,1)$) based on the distribution F: $$q_{n}(t) = \{\sqrt{n}[F^{-1}_{n}(t)-F^{-1}(t)]\}.$$ In this case, $F^{-1}$ is the (generalized) inverse of ...
1
vote
2answers
36 views

How to efficiently simulate successes of several trials if probabilities are inhomogeneous

If I'm doing a simulation with $n$ trials, each with probability $p$, a quick way to select the successful trials is to choose a binomially distributed random number. Then randomly choose that many ...
1
vote
0answers
39 views

Distribution of $(\sup_{0\leq s\leq t} W_s -W_t)$

I am interest in the law of the $(\sup_{0\leq s\leq t} W_s -W_t)$ where $W$ is a standard brownian motion. I know that $M_t:=\sup_{0\leq s\leq t} W_s \overset{\mathcal L}{=} |W_t |$ so its density ...
0
votes
1answer
41 views

Proving that Doob's martingale is a martingale

I'm working on my first ever proof that a stochastic process is a martingale, and I'm a bit confused. Is there a "standard machine" for these proofs? To be more specific, I am trying to show that if ...
0
votes
0answers
17 views

Inverse Bessel Process

Is there any reference on this process? For example, analytical derivations for the hitting times, density, etc? Im studying local martingales and am interested in the density of hitting times for ...
0
votes
0answers
21 views

Distribution of points in a homogeneous PPP

PPP holds some important properties. However, my question is whether the positions of the points in a homogeneous PPP are independent? Equivalently, are the points in a homogeneous PPP distributed ...
1
vote
0answers
26 views

Question about Lebesgue Dominated Convergence Theorem involving a Markov Time / Stopping Time

I am trying to understand the proof of the following lemma: Let $W$ be an arbitrary random variable satisfying $\mathbb{E}[|W|] < \infty$, and let $T$ be a Markov time (or stopping time) for which ...
1
vote
0answers
45 views

Couple/Compare two stochastic processes and prove an intuitive proposition

Consider a stochastic process (denoted $X$) $X_0, X_1, X_2, \ldots$ (not necessarily a Markov Chain) over state space $\{0, 1, \cdots, n \}$. The transition probabilities are $$P(X_{i+1} = 1 \mid ...
1
vote
0answers
70 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
2
votes
0answers
21 views

Stability of non-autonomous stochastic differential equation

I'm looking for a good reference or insight to under what conditions can I prove stability (or instability) for the following general n-dimensional non-autonomous stochastic differential equation: ...
0
votes
1answer
29 views

Product topology and uniform topology on C[0,T]

Is the product topology on $\mathbb{R}^{[0,T]}$ restricted to $C[0,T]$ (T finite) the same as the topology induced by the uniform norm on $C[0,T]$? I am curious because I saw a claim on wiki saying ...
1
vote
1answer
23 views

What is the difference between “filtration for a Brownian motion” and “filtration generated by a Brownian motion”?

I'm reading Shreve's book "Stochastic Calculus for Finance: Vol II". In 5.3.1, after the Theorem 5.3.1 (Martingale representation, one dimension), Shreve explains: "The assumption that the filtration ...
3
votes
1answer
29 views

Natural Filtration and Sigma-Field Generated by path function

Suppose we have a continuous real-valued stochastic process $X=(X_t;t\geq 0)$ defined on a probability space $(\Omega,F,P)$. Usually one defined the filtration to be $F_t=\sigma(X_s;s\leq t)$. But on ...
1
vote
0answers
50 views

Probability of going to the origin in a random walk

Been given this as practice for my Stochastic Processes course. I'm fairly new to the concept, so I haven't been exposed to a general method. Any hints/tips for the following? A gambler plays a ...
0
votes
1answer
27 views

Covariance of time series.

Let $\varepsilon_n \sim \textrm{WN}(0,\tau^2) $ be the white noise. Calculate $\textrm{Cov}(X_n, X_{n+k})$, where $X_n = \varepsilon_n(\varepsilon_n - \varepsilon_{n-1})$. Can anybody help? I've just ...
2
votes
0answers
22 views

Convergence of sum of exp. decaying pdf // When does L^2 convergence imply a.s. convergence?

The problem: Let $X_t^i$ $(i \in Z)$ be integer valued random variables on the same probability space. Let $m: Z \rightarrow R $ be a symmetric probability density on the integers which has ...
2
votes
0answers
6 views

Reference request random time changes representations and weak convergence

I was reading the Kurtz's book 'Markov Processes: Characterization and Convergence' and I need to prove a similar theorem as theorem 1.5 on chapter 6 of that book, that basically states that if $Y$ is ...
0
votes
0answers
29 views

Convergence in finite-dimensional distributions of stochastic processes and random measures

Motivation: Let $(\Omega, \mathscr{A}, P)$ be a probability space and consider the measurable spaces $D := D[0,\infty)$ of cadlag functions and $M := M[0,\infty)$ of Radon measures on $[0,\infty)$. ...